The Wikipedia article on Egyptian mathematics suggests several sources from which you could glean further insight and specific details.

Even in the modern world, there is a sharp difference between the way mathematics is perceived by mathematicians, laypeople, and mathematics-using specialists (scientists and engineers). I mean, comparing the answers of questions like, "What is the goal of mathematics," "What is mathematics," "How does one learn/become better at mathematics," "Where is mathematics, as a human discipline, going." I think these same questions could be posed to and answered by a hypothetical Ancient Egyptian. Better yet, contrasted between ancient ancient Egyptians, ancient Egyptians, and merely old Egyptians.

All of this is contingent on identifying a paradigm-shift in Egyptian mathematical thinking. A natural analog for the ancient Greek might have been the attempts at making a formal study of irrational numbers (provided that this attempt was made by the Greeks in any mainstream capacity...). Perhaps a better example would be the resetting of classical mathematics in terms of set theory by the Bourbaki group during the early 20th century (whether this is a "paradigm shift" or merely the introduction of "better pedagogy" is not my place to qualify). If you could identify a point in time when Egyptians starting doing something mathematically new/different (and perhaps the resulting civil improvements/changes) it would certainly be interesting.

Could you tell us some more about this problem? Laguerre polynomials are orthonormal when given the inner-product <L_n,L_m> = int L_n L_m e-x from 0 to infinity, so getting a delta on the right-hand side is reasonable; but the f(n) is somewhat concerning, since you could switch the two polynomials in the integrand with no bearing on the evaluation of the integral (unless w(x) has some other dependence on n,m we don't know about).

That was my first thought as well. One would think of normalizing the inner product of L_n and L_m against f(n) through the weight w(x).

However, if the integrand does indeed represent some kind of weighted inner-product on Laguerre polynomials, then the dependence of n in f on the right-hand-side will certainly be impossible since inner-products are symmetric.

Using the built-in functions was my go-to solution, but the code deals fundamentally with recasting these operators on different manifolds (while still operating on vectors in R3 instead of local coordinates). I threw in the original definitions explicitly (and as you noticed, some redundant definitions when the two cases coincide) in order to help the code document itself.

You're totally right, though. These are exactly the standard Div and Grad. I still might go back to using them in a later version.

Thanks again for showing me that you can recast the result as a pure function.

That's basically the issue, although with a caveat: if I just want to evaluate the output numerically, I can use replacement rules and Mathematica will just poop out what I need. The problem comes up when I need to symbolically differentiate the output.

I have only a few things that looks like Qx[{x,y,z}].F[x,y,z,t], but I also have several that look like Qx[{x,y,z}].grad[f][x,y,z,t], or other compositions of that form (Qx with some operator, then hit f). I could write out Qx[{x,y,z}].F[x,y,z,t] explicitly and define the appropriate function, since I know F = {fx,fy,fz} ahead of time, however a general solution would be deeply appreciated.